Phase-Field Approximation of the Willmore Flow

Fei, Mingwen; Liu, Yuning

doi:10.1007/s00205-021-01678-9

Cited by 8 publications

(10 citation statements)

References 28 publications

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“…For the estimate on Err M Cξ , we have to work a bit more. We begin by adding zeroes and using (11) to obtain…”

Section: Strategy Of the Proofmentioning

confidence: 99%

“…For several important structural classes of potentials W , such a rigorous analysis has long been available for the Allen-Cahn equation: For instance, for the scalar Allen-Cahn equation with twowell potential W -that is, for (1) with N = 2 -the convergence towards (twophase) mean curvature flow in the limit ε → 0 has been established by De Mottoni and Schatzman [9], Bronsard and Kohn [5], Chen [7], Ilmanen [16], and Evans, Soner, and Souganidis [10] in the context of three different notions of solutions to mean curvature flow (namely, strong solutions, Brakke solutions, respectively viscosity solutions). In such two-phase situations, sharp-interface limits have also been established for more complex phase-field models [8,3,1,11,2], typically based on an approach that relies on matched asymptotic expansions and a stability analysis of the PDE linearized around a transition profile. Beyond the case of twowell potentials, results have been much more scarce.…”

Section: Introductionmentioning

confidence: 99%

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Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

Fischer¹,

Marveggio²

2022

Preprint

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Phase-field models such as the Allen-Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen-Cahn equation with a potential with N ≥ 3 distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow.In the present work, we give a rigorous proof for this statement in two and three ambient dimensions and for a suitable class of potentials: As long as a strong solution to multiphase mean curvature flow exists, solutions to the vectorial Allen-Cahn equation with well-prepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter ε 0. We even establish the rate of convergence O(ε 1/2 ). Our approach is based on the gradient flow structure of the Allen-Cahn equation and its limiting motion: Building on the recent concept of "gradient flow calibrations" for multiphase mean curvature flow, we introduce a notion of relative entropy for the vectorial Allen-Cahn equation with multi-well potential. This enables us to overcome the limitations of other approaches, e. g. avoiding the need for a stability analysis of the Allen-Cahn operator or additional convergence hypotheses for the energy at positive times.

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“…For the estimate on Err M Cξ , we have to work a bit more. We begin by adding zeroes and using (11) to obtain…”

Section: Strategy Of the Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

Fischer¹,

Marveggio²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…This however does not give a general convergence proof, since the assumed properties need to be verified for a phase field approximation. Complete convergence proofs based on asymptotic expansion techniques are known for the standard diffuse approximation of mean curvature and Willmore flow, see [60] and [27].…”

Section: Convergence Towards the Mean Curvature Flow And Willmore Flo...mentioning

confidence: 99%

“…Corresponding diffuse approximations of the Willmore flow have been introduced by [24] and have been justified by formal asymptotic expansions in [25], see also [26]. In a recent article [27] Fei and Liu prove the convergence of diffuse approximations to the Willmore flow for well-prepared initial data, as long as the smooth limit flow exists. Quite a number of numerical schemes for the simulation of the Willmore flow have been proposed.…”

Section: Introductionmentioning

confidence: 99%

"Gradient-free" diffuse approximations of the Willmore functional and Willmore flow

Dabrock,

Knüttel,

Röger

2022

Preprint

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We introduce new diffuse approximations of the Willmore functional and the Willmore flow. They are based on a corresponding approximation of the perimeter that has been studied by Amstutz-van Goethem [Interfaces Free Bound. 14 (2012)]. We identify the candidate for the Γ-convergence, prove the Γ-limsup statement and justify the convergence to the Willmore flow by an asymptotic expansion. Furthermore, we present numerical simulations that are based on the new approximation.

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“…For that we will use the method of formal asymptotic analysis. The techniques we employ are similar to those applied for the asymptotic analysis of related phase field models like [4], the Stokes-Allen-Cahn system in [1], or the Willmore 2 -flow [9]. Another related asymptotic analysis is that of [23] for minimisers of the Canham-Helfrich energy.…”

Section: Introductionmentioning

confidence: 99%

Sharp interface analysis of a diffuse interface model for cell blebbing with linker dynamics

Nöldner¹,

Burger²,

Garcke³

2023

Preprint

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We investigate the convergence of solutions of a recently proposed diffuse interface/phase field model for cell blebbing by means of matched asymptotic expansions. It is a biological phenomenon that increasingly attracts attention by both experimental and theoretical communities. Key to understanding the process of cell blebbing mechanically are proteins that link the cell cortex and the cell membrane. Another important model component is the bending energy of the cell membrane and cell cortex which accounts for differential equations up to sixth order. Both aspects pose interesting mathematical challenges that will be addressed in this work like showing non-singularity formation for the pressure at boundary layers, deriving equations for asymptotic series coefficients of uncommonly high order, and dealing with a highly coupled system of equations.

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Phase-Field Approximation of the Willmore Flow

Cited by 8 publications

References 28 publications

Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

"Gradient-free" diffuse approximations of the Willmore functional and Willmore flow

Sharp interface analysis of a diffuse interface model for cell blebbing with linker dynamics

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